Topic 10.4 Electric Potential Energy: calculate the electric potential energy of a system of point charges and relate it to work done.

What this topic is asking

The College Board (Topic 10.4) wants you to calculate the electric potential energy of a system of point charges, $U = \dfrac{k q_1 q_2}{r}$, interpret its sign, and connect it through energy conservation to the work done and the kinetic energy of moving charges.

What electric potential energy is

Potential energy is energy of configuration: it depends on where the charges sit, not on how they got there. The reference point is infinity, where $U = 0$, so $U$ at separation $r$ is the work done against (or by) the electric force to bring them that close. Because the force is conservative (like gravity), this work depends only on the start and end positions, making $U$ a well-defined function of the arrangement.

The role of sign and distance

The sign tells the physics story. Two like charges resist being pushed together, storing positive energy that they will release as kinetic energy if let go (they fly apart). Two unlike charges attract, so assembling them releases energy and $U$ is negative; separating them requires work, raising $U$ toward zero. A frequent exam point is that pulling unlike charges apart increases $U$ (from a large negative value toward zero), even though it does not become positive.

Many charges and energy conservation

For a system of several charges, the total potential energy is the sum over all distinct pairs: compute $\dfrac{k q_i q_j}{r_{ij}}$ for each pair and add them (with signs). The deeper use of $U$ is through conservation of energy: the electric force is conservative, so the total mechanical energy $KE + U$ is constant for charges moving under electric forces alone. A charge released from rest speeds up as it moves to lower potential energy, converting $U$ into $KE$ exactly as a falling mass converts gravitational potential energy into kinetic energy. This is the standard problem type: set the lost potential energy equal to the gained kinetic energy, $\Delta KE = -\Delta U$, and solve for the speed. The strategic thread is that potential energy connects the force picture (Topics 10.1, 10.3) to the energy picture, and dividing $U$ by the charge gives the electric potential (voltage) of Topic 10.5, the energy-per-charge quantity that runs every circuit.

Try this

Q1. State the sign of the electric potential energy of two negative charges. [1 point]

• Cue. Positive (the product of two negatives is positive).

Q2. A charge moves to a region of lower electric potential energy. State what happens to its kinetic energy. [1 point]

• Cue. It increases (energy is conserved, so lost potential energy becomes kinetic energy).